This 100 Hz first-order combination tone then interacts with both notes of the interval to produce second-order combination tones of 200 ( 300 100 ) and 100 ( 200 100 ) Hz and all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300.
22.
He designed and described numerous non-octave temperaments ), many based on combination tones, including the Bohlen Pierce scale in 1972 ( independently discovered by John R . Pierce in 1984, also a microwave electronics and communications engineer, six years later and Kees van Prooijen in 1978 ), the A12 scale, and the 833 cents scale.
23.
There are two types of combination tones : "'sum tones "'whose frequencies are found by adding the frequencies of the real tones and "'difference tones "'whose frequencies are the difference between the frequencies of the real tones . " Combination tones are heard when two pure tones ( i . e ., tones produced by simple harmonic sound waves having no overtones ), differing in frequency by about 50 cycles per second or more, sound together at sufficient intensity ."
24.
There are two types of combination tones : "'sum tones "'whose frequencies are found by adding the frequencies of the real tones and "'difference tones "'whose frequencies are the difference between the frequencies of the real tones . " Combination tones are heard when two pure tones ( i . e ., tones produced by simple harmonic sound waves having no overtones ), differing in frequency by about 50 cycles per second or more, sound together at sufficient intensity ."
25.
For example, 220 440 Hz produce combination tones at 220 Hz and 660 Hz . 660 Hz is a perfect fifth ( 3 : 2 ) above 440 Hz, and produce combination tones at 220 Hz and 1, 100 Hz . 1, 100 Hz is a major sixth ( 5 : 3 ) above 660 Hz, and produce combination tones at 440 Hz and 1, 760 Hz . 1100 Hz and 1760 Hz are a minor sixth ( 8 : 5 ), and so on . " It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent ."
26.
For example, 220 440 Hz produce combination tones at 220 Hz and 660 Hz . 660 Hz is a perfect fifth ( 3 : 2 ) above 440 Hz, and produce combination tones at 220 Hz and 1, 100 Hz . 1, 100 Hz is a major sixth ( 5 : 3 ) above 660 Hz, and produce combination tones at 440 Hz and 1, 760 Hz . 1100 Hz and 1760 Hz are a minor sixth ( 8 : 5 ), and so on . " It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent ."
27.
For example, 220 440 Hz produce combination tones at 220 Hz and 660 Hz . 660 Hz is a perfect fifth ( 3 : 2 ) above 440 Hz, and produce combination tones at 220 Hz and 1, 100 Hz . 1, 100 Hz is a major sixth ( 5 : 3 ) above 660 Hz, and produce combination tones at 440 Hz and 1, 760 Hz . 1100 Hz and 1760 Hz are a minor sixth ( 8 : 5 ), and so on . " It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent ."
28.
In his 1937 writing " Unterweisung im Tonsatz " ( " The Craft of Musical Composition ", ), he wrote that " notes have a family of relationships, that are the bindings of tonality, in which the ranking of intervals is unambiguous, " so much so, indeed, that in the art of triadic composition " . . . the musician is bound by this, as the painter to his primary colours, the architect to the three dimensions . " He lined up the harmonic and melodic aspects of music in a row in which the octave ranks first, then the fifth and the third, and then the fourth . " The strongest and most unique harmonic interval after the octave is the fifth, the prettiest nevertheless is the third by right of the chordal effects of its Combination tones ."
